i would have reraised the flop too. A lot of outs (nut flush + trips + twopairs= wow in heads-up). You already have top pair topkicker, which is the hand you wanted to find playing AK preflop, with a lot of outs in bonus!

Moreover: i never respect minraise, especially when i have a monstruous hand in headsup.

You practically have the nuts on the flop. Well not really but great equity vs any hand. 3betting flop to ~$9 and pushing turn is given for me. You make more money when you use your fold equity and they fold. Plus it eliminates hard decision on later streets.

Since there isn´t too many cards that could hurt you I would go for second choice, since his turn bet looked weak+ I might be still ahead against flushdraws+ I have fold equity+ against flush I have 7 outs and against set I have 9 outs and against two pair hand I have 12 outs.